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This paper aims to control the in-plane wave propagation characteristics of two-dimensional functionally graded periodic grid structures by adding local resonators made of the spring oscillator. The in-plane wave dispersion relations of the periodic grid structures with local resonators are calculated by spectral element method combined with Bloch theorem. Meanwhile, the effectiveness of the band gap calculation method is verified by the vibration transmission of the finite-length grid structures calculated based on spectral element method and finite element method. According to the spectral element method, the spectral stiffness matrix of the functionally graded beam element is established at first, then assembled with the additional stiffness matrix of the spring oscillator subsystem to obtain the complete spectral stiffness matrix of the functionally graded oscillator coupled beam. Finally, the whole stiffness matrix of grid structures with local resonators can be obtained by the coordinate transformation matrix, to form the in-plane wave dispersion relations equation based on the Bloch periodic boundary conditions. In addition, the effects of structural and material parameters on the in-plane wave propagation characteristics are analyzed, which can be applied to the vibration reduction design of periodic grid structures.

Band gap characteristics of two-dimensional functionally graded periodic grid structures with local resonators